How to Improve Physics/Math Understanding as an Undergraduate?

Hello, everyone! I'm a sophomore Physics and Math major from Temple University. Currently my GPA is 3.91 and I'm currently in Differential Equations, multivariable calculus, and Physics 2 (introductory E&M type of class basically). My problem is that, although I've been doing great grade-wise in all of my classes, I feel like I'm not truly learning the material. I can do all the computations and such without any problem really, but the proofs and underlying concepts seem to go right over my head. I feel like this is going to cause me a ton of trouble after this semester when I get to the upper level courses (advanced calculus, classical mechanics, etc) and I want to lessen that trouble if I can in some way. Is there any efficient way to increase my understanding of the material outside of class? I want to eventually land an REU and go to graduate school, and I'm not sure if I'll be able to do either without really understanding everything I'm learning in these fundamental courses.

Also, as a little side question, does anyone recommend a certain way to learn to program? I've read that it's necessary to have at least a slight aptitude with it but my school only requires one course in programming for my major, and I feel like that won't be enough at all, so I plan to try and self-learn a bit over the summer.

As for programming, there is only one way to get good at it. Pick your favorite programming language (Lisp, Python, Java, C++) and start programming. You might want to set yourself a goal of a program that you make and then make it. For example, you might want to make a program that simulates the motion and collisions of a few particles. The more you program, the better you'll be.

As for programming, there is only one way to get good at it. Pick your favorite programming language (Lisp, Python, Java, C++) and start programming. You might want to set yourself a goal of a program that you make and then make it. For example, you might want to make a program that simulates the motion and collisions of a few particles. The more you program, the better you'll be.

Thank you! It's not so much that I'm struggling with proofs yet, since I haven't even had a class that requires them yet outside of the little bit of the precise definition of a limit that I did in Calculus 1 with Epsilons/Deltas which was easy, but I have to jump straight from no proofs to advanced calculus in the fall for next year, and I'm just worried that if I don't completely understand everything I'm learning now then it will be a semester from hell. I will definitely look into that book.

I already have Spivak's Calculus on Manifolds, Hardy's A Course of Pure Mathematics, Buck's Advanced Calculus 3E, and Apostol's Mathematical Analysis, but I haven't looked at them yet. Are they up to par with that Velleman's book?

I already have Spivak's Calculus on Manifolds, Hardy's A Course of Pure Mathematics, Buck's Advanced Calculus 3E, and Apostol's Mathematical Analysis, but I haven't looked at them yet. Are they up to par with that Velleman's book?

Try to read those books. If you can read them without problem, then you have no real problem with proofs. (it are quite difficult books)

Hey I'm in the exactly the same situation that you are(I'm taking the same classes). Honestly the math part of physics 2 is easy. The hard part is setting up the problem. I know it's very tempting to look at the solutions when solving problems, try to avoid that as much as possible. Do the challenge problems in your textbook without any help at all. If you want I can give you the hw my professor assigns us; if you can solve those without help, then you understand it enough.

You got to let your intuition do some work. When your're solving some of these problems you might not have any clue where to start, but all of a sudden this moment of realization comes and it feels good. Doing 100's of simple math calculations won't teach you that.

You're doing great grade wise so stop worrying! You're "learning the material"! To want to "truly learn the material" is probably evidence of "existential anxiety" rather than of anything "going over your head". You obviously understand the material "well enough". If you are on the Quixotic quest for "fundamental meaning" change to philosophy...

You're doing great grade wise so stop worrying! You're "learning the material"! To want to "truly learn the material" is probably evidence of "existential anxiety" rather than of anything "going over your head". You obviously understand the material "well enough". If you are on the Quixotic quest for "fundamental meaning" change to philosophy...

I appreciate the kind words, but I'm just afraid that my grades are only because my classes haven't been rigorous enough. Hope you're right though!

Another quick question: Currently, I really want to study abroad, so I'm planning on applying for the Mathematics in Budapest program for the Fall of my senior year; however, if I do this, I'd need to really take a difficult semester on the Fall beforehand so I can accomplish enough to graduate during my senior spring. Is that an awful idea? If I do it as I'm planning to, my course load will look like:

Advanced Calculus I
Analytical Mechanics
Electricity and Magnetism I
Computing for Scientists
Thermodynamics and Kinetic Theory

I appreciate the kind words, but I'm just afraid that my grades are only because my classes haven't been rigorous enough. Hope you're right though!

Another quick question: Currently, I really want to study abroad, so I'm planning on applying for the Mathematics in Budapest program for the Fall of my senior year; however, if I do this, I'd need to really take a difficult semester on the Fall beforehand so I can accomplish enough to graduate during my senior spring. Is that an awful idea? If I do it as I'm planning to, my course load will look like:

Advanced Calculus I
Analytical Mechanics
Electricity and Magnetism I
Computing for Scientists
Thermodynamics and Kinetic Theory

Basically I'm asking is this a manageable course load?

I was asking a similar question to yours a few months ago, if you take a look at my thread history. The thing I noticed, even though I still don't truly understand everything is that at the introductory level, you are not expected to know every derivation, every proof and the concepts at a fundamental level. They're more geared towards getting you familiar with them and getting you used to doing computations with them (hence one can pull off an A without having "truly" learnt the material). Once you go on to more advanced courses, they will use the concepts in the introductory courses frequently and at a more fundamental level and if you put enough time trying to understand/studying those harder courses, the simpler concepts will also start making a lot more intuitive sense. In general, once you've worked with something for a long enough time, even if you didn't fully understand it when you first learned it, you will start seeing the "why's".

As for that courseload, it seems manageable but it also depends on you. If you're willing to spend the majority of your time studying/doing homework, then you should be fine with the courseload.